Hmm... is there a notation called 7!!! where it's 7 x 4 x 1?

Prof. Loh said here that 7!! = 7 x 5 x 3 x 1.
Following that logic, is there a notation called 7!!! where it's 7 x 4 x 1?

Would (7!)! be (7 x 6 x 5 x 4 x 3 x 2 x 1)! ? Also, I remember in a math contest it asked something like "What is 4!?" and I was really confused because I thought it was just an exciting question and that the answer was 4 because 4 is 4.

Lol these notations are weird

What would 8!! be? 8 x 6 x 4 x 2?

I think it would actually be also 7x5x3x1

btw guys i just changed my username from da parasite to bulba_bulbasaur

@divinedolphin
Yeah Prof Loh explained in the vid I think.
Also when I was doing the four 4's problem (arrange four 4's and add operations or "smush"  Prof Loh word  them together to get any number from 150) I started using !! like mad  around 3/4 of the time (and I was the only person in the class using it frequently)